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already filled with nk photons on average is 1 + nk greater than that of emitting into an unpopulated mode. 4 With these ideas, we are already in a position to write down a "back of the envelope" description of a laser. In this "hand-waving" description, we will regard the various quantum-mechanical observables as if they were classical. Laser cavities are usually open, so some of the photons emitted can actually go into free space. Let us call (3 the fraction of emission into the lasing mode. Then if we call n the number of photons in the lasing mode, N the population of the upper lasing level, and I'll the decay rate of the upper lasing level to the lower lasing level, the rate at which stimulated emission photons will increase the population of the lasing mode is5 f3'YIiNn. Cavity damping will tend to decrease this population at a rate of -Kn, where K- 1 is the photon lifetime in the lasing mode. Absorption of photons by the population in the lower lasing level (making then a transition to the upper level) will also tend to decrease n. But if the lower lasing level is rapidly emptied into lower-lying levels that do not tak~: part in the laser, there will be no absorption. For simplicity, let us assume that this is the case. Then it is only the population N of the upper lasing level that appears in the laser equations. Last but not least, there is the spontaneous emission contribution to the population of the lasing mode, R"p = f3'YIiN. Putting all this together, we have
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To complete our description, we must write down the rate equation for N. The spontaneous (into all modes now) and stimulated emission terms will, of course, tend to decrease N. To keep the laser going, we must feed energy in it. This is done by pumping the population in the upper lasing level with some rate A. Thus, (8.7) The lifetime of molecular and atomic energy levels involved in optical transitions is often so short that even a moderate cavity Q value is already large enough to make K much smaller than 1'11' In that case, the relaxation dynamics of N will be much faster than that of nand N will "follow n adiabatically." This is called adiabatic elimination. It can be derived in the following way. If we multiply (8.7) by exp(')'11 t) and notice that 'YIIN exp('Yllt) = Ndexp('Yllt)/dt, we can rewrite (8.7) as
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d dt (Ne'Yllt) == (A - 'Y1lf3Nn)e'Ylit.
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4Por fermions we have the anticornmutation relation {bk,bt} == bkbt + btbk = 1 rather than the commutation relation above. As a consequence, the probability of emission into a given mode k turns out to be proportional to Ibt I"') 12 = 1 - fib Sl) that the probability of emission actually decreases if the mode is not absolutely unpopulated. Notice that the average number of fermions on mode k. fik. cannot be larger than 1; otherwise. the square modulus of btl"') becomes negative. 5The stimulated emission rate is that part of the emission rate that exists only when the mode is already populated. Hence. it is the fik part of fik + 1 derived in the preceding paragraphs. The "I" part is the spontaneous emission contribution, which is always present.
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Integrating this equation from the initial time unpopulated, to t, we get
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= [too dt' {A -111,8N(t')n(t')} e-(t-t'hll.
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As 111 K, the integrand above is dominated by the exponential. Using the steepest descent method, we can approximate this integral by
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~ {A -111,8N(t)n(t)}
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dt' e-(t-t'hll
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A -111,8N(t)n(t) . 111
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Solving for N yields
N= AfTll 1 +,8n'
which shows that adiabatic elimination of N is equivalent to setting
to zero in
In ordinary lasers, ,8 is often much smaller than 1. A ReNe 633-nm laser, for example, has ,8 ~ 10-8 , and an index-guided edge-emitting semiconductor laser typically has ,8 ~ 10- 5 [654]. Thus, for n 1/,8, we can expand the denominator on the right-hand side of (8.11) to first order in ,8n. Substituting the result in (8.6), and neglecting the spontaneous emission contribution Rsp , we get the following differential equation for n: (8.12) Calling A,8 - K and A,82 on the right-hand side of (8.12) a/2 and r/ /2, respectively, we can recognize here the van der Pol oscillator in the slowly varying envelope and rotating-wave approximations. But this is a van der Pol oscillator that oscillates at optical rather than radio frequencies, an increase of seven orders of magnitude: from hundreds of megahertz to hundreds of terahertz. We mentioned earlier that it is noise that knocks the van der Pol oscillator out of the unstable = 0 steady state when we switch on the circuit. Without noise, = 0 some other it would never start oscillating unless we perturbed it out of way. But we have otherwise neglected noise in our analysis so far. Although in electronics neglecting noise is not so bad, in lasers (quantum electronics) it is a very poor approximation. This is because at radio frequencies, noise is of classical origin. In that regime, the energy of an electromagnetic quantum (photon) is much smaller than the energy of thermal fluctuations (kT) at room temperature. As classical noise can be reduced, we can get it to a level where it is negligible. At optical frequencies, however, quantum noise is no longer negligible. As quantum noise cannot be eliminated,6 we just have to live with it. In the laser it plays an important
6Quantum noise can be reduced for one observable only at the expense of increasing it for the complementary observable, as in squeezed light.